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.cm-s-inner .cm-qualifier { color: rgb(85, 85, 85); }
.cm-s-inner .cm-builtin { color: rgb(51, 0, 170); }
.cm-s-inner .cm-bracket { color: rgb(153, 153, 119); }
.cm-s-inner .cm-tag { color: rgb(17, 119, 0); }
.cm-s-inner .cm-attribute { color: rgb(0, 0, 204); }
.cm-s-inner .cm-header, .cm-s-inner.cm-header { color: rgb(0, 0, 255); }
.cm-s-inner .cm-quote, .cm-s-inner.cm-quote { color: rgb(0, 153, 0); }
.cm-s-inner .cm-hr, .cm-s-inner.cm-hr { color: rgb(153, 153, 153); }
.cm-s-inner .cm-link, .cm-s-inner.cm-link { color: rgb(0, 0, 204); }
.cm-negative { color: rgb(221, 68, 68); }
.cm-positive { color: rgb(34, 153, 34); }
.cm-header, .cm-strong { font-weight: 700; }
.cm-del { text-decoration: line-through; }
.cm-em { font-style: italic; }
.cm-link { text-decoration: underline; }
.cm-error { color: red; }
.cm-invalidchar { color: red; }
.cm-constant { color: rgb(38, 139, 210); }
.cm-defined { color: rgb(181, 137, 0); }
div.CodeMirror span.CodeMirror-matchingbracket { color: rgb(0, 255, 0); }
div.CodeMirror span.CodeMirror-nonmatchingbracket { color: rgb(255, 34, 34); }
.cm-s-inner .CodeMirror-activeline-background { background: inherit; }
.CodeMirror { position: relative; overflow: hidden; }
.CodeMirror-scroll { height: 100%; outline: 0px; position: relative; box-sizing: content-box; background: inherit; }
.CodeMirror-sizer { position: relative; }
.CodeMirror-gutter-filler, .CodeMirror-hscrollbar, .CodeMirror-scrollbar-filler, .CodeMirror-vscrollbar { position: absolute; z-index: 6; display: none; }
.CodeMirror-vscrollbar { right: 0px; top: 0px; overflow: hidden; }
.CodeMirror-hscrollbar { bottom: 0px; left: 0px; overflow: hidden; }
.CodeMirror-scrollbar-filler { right: 0px; bottom: 0px; }
.CodeMirror-gutter-filler { left: 0px; bottom: 0px; }
.CodeMirror-gutters { position: absolute; left: 0px; top: 0px; padding-bottom: 30px; z-index: 3; }
.CodeMirror-gutter { white-space: normal; height: 100%; box-sizing: content-box; padding-bottom: 30px; margin-bottom: -32px; display: inline-block; }
.CodeMirror-gutter-wrapper { position: absolute; z-index: 4; background: 0px 0px !important; border: none !important; }
.CodeMirror-gutter-background { position: absolute; top: 0px; bottom: 0px; z-index: 4; }
.CodeMirror-gutter-elt { position: absolute; cursor: default; z-index: 4; }
.CodeMirror-lines { cursor: text; }
.CodeMirror pre { border-radius: 0px; border-width: 0px; background: 0px 0px; font-family: inherit; font-size: inherit; margin: 0px; white-space: pre; overflow-wrap: normal; color: inherit; z-index: 2; position: relative; overflow: visible; }
.CodeMirror-wrap pre { overflow-wrap: break-word; white-space: pre-wrap; word-break: normal; }
.CodeMirror-code pre { border-right: 30px solid transparent; width: fit-content; }
.CodeMirror-wrap .CodeMirror-code pre { border-right: none; width: auto; }
.CodeMirror-linebackground { position: absolute; left: 0px; right: 0px; top: 0px; bottom: 0px; z-index: 0; }
.CodeMirror-linewidget { position: relative; z-index: 2; overflow: auto; }
.CodeMirror-wrap .CodeMirror-scroll { overflow-x: hidden; }
.CodeMirror-measure { position: absolute; width: 100%; height: 0px; overflow: hidden; visibility: hidden; }
.CodeMirror-measure pre { position: static; }
.CodeMirror div.CodeMirror-cursor { position: absolute; visibility: hidden; border-right: none; width: 0px; }
.CodeMirror div.CodeMirror-cursor { visibility: hidden; }
.CodeMirror-focused div.CodeMirror-cursor { visibility: inherit; }
.cm-searching { background: rgba(255, 255, 0, 0.4); }
@media print {
  .CodeMirror div.CodeMirror-cursor { visibility: hidden; }
}


/* Flowchart variables */
/* Sequence Diagram variables */
/* Gantt chart variables */
/* state colors */
.label {
  
  color: #333; }

.label text {
  fill: #333; }

.node rect,
.node circle,
.node ellipse,
.node polygon {
  fill: #BDD5EA;
  stroke: #9370DB;
  stroke-width: 1px; }

.node .label {
  text-align: center; }

.node.clickable {
  cursor: pointer; }

.arrowheadPath {
  fill: lightgrey; }

.edgePath .path {
  stroke: lightgrey;
  stroke-width: 1.5px; }

.edgeLabel {
  background-color: #e8e8e8;
  text-align: center; }

.cluster rect {
  fill: #6D6D65;
  stroke: rgba(255, 255, 255, 0.25);
  stroke-width: 1px; }

.cluster text {
  fill: #F9FFFE; }

div.mermaidTooltip {
  position: absolute;
  text-align: center;
  max-width: 200px;
  padding: 2px;
  
  font-size: 12px;
  background: #6D6D65;
  border: 1px solid rgba(255, 255, 255, 0.25);
  border-radius: 2px;
  pointer-events: none;
  z-index: 100; }

.actor {
  stroke: #81B1DB;
  fill: #BDD5EA; }

text.actor {
  fill: black;
  stroke: none; }

.actor-line {
  stroke: lightgrey; }

.messageLine0 {
  stroke-width: 1.5;
  stroke-dasharray: '2 2';
  stroke: lightgrey; }

.messageLine1 {
  stroke-width: 1.5;
  stroke-dasharray: '2 2';
  stroke: lightgrey; }

#arrowhead {
  fill: lightgrey; }

.sequenceNumber {
  fill: white; }

#sequencenumber {
  fill: lightgrey; }

#crosshead path {
  fill: lightgrey !important;
  stroke: lightgrey !important; }

.messageText {
  fill: lightgrey;
  stroke: none; }

.labelBox {
  stroke: #81B1DB;
  fill: #BDD5EA; }

.labelText {
  fill: #323D47;
  stroke: none; }

.loopText {
  fill: lightgrey;
  stroke: none; }

.loopLine {
  stroke-width: 2;
  stroke-dasharray: '2 2';
  stroke: #81B1DB; }

.note {
  stroke: rgba(255, 255, 255, 0.25);
  fill: #fff5ad; }

.noteText {
  fill: black;
  stroke: none;
  
  font-size: 14px; }

.activation0 {
  fill: #f4f4f4;
  stroke: #666; }

.activation1 {
  fill: #f4f4f4;
  stroke: #666; }

.activation2 {
  fill: #f4f4f4;
  stroke: #666; }

/** Section styling */
.section {
  stroke: none;
  opacity: 0.2; }

.section0 {
  fill: rgba(255, 255, 255, 0.3); }

.section2 {
  fill: #EAE8B9; }

.section1,
.section3 {
  fill: white;
  opacity: 0.2; }

.sectionTitle0 {
  fill: #F9FFFE; }

.sectionTitle1 {
  fill: #F9FFFE; }

.sectionTitle2 {
  fill: #F9FFFE; }

.sectionTitle3 {
  fill: #F9FFFE; }

.sectionTitle {
  text-anchor: start;
  font-size: 11px;
  text-height: 14px;
   }

/* Grid and axis */
.grid .tick {
  stroke: lightgrey;
  opacity: 0.3;
  shape-rendering: crispEdges; }

.grid path {
  stroke-width: 0; }

/* Today line */
.today {
  fill: none;
  stroke: #DB5757;
  stroke-width: 2px; }

/* Task styling */
/* Default task */
.task {
  stroke-width: 2; }

.taskText {
  text-anchor: middle;
   }

.taskText:not([font-size]) {
  font-size: 11px; }

.taskTextOutsideRight {
  fill: #323D47;
  text-anchor: start;
  font-size: 11px;
   }

.taskTextOutsideLeft {
  fill: #323D47;
  text-anchor: end;
  font-size: 11px; }

/* Special case clickable */
.task.clickable {
  cursor: pointer; }

.taskText.clickable {
  cursor: pointer;
  fill: #003163 !important;
  font-weight: bold; }

.taskTextOutsideLeft.clickable {
  cursor: pointer;
  fill: #003163 !important;
  font-weight: bold; }

.taskTextOutsideRight.clickable {
  cursor: pointer;
  fill: #003163 !important;
  font-weight: bold; }

/* Specific task settings for the sections*/
.taskText0,
.taskText1,
.taskText2,
.taskText3 {
  fill: #323D47; }

.task0,
.task1,
.task2,
.task3 {
  fill: #BDD5EA;
  stroke: rgba(255, 255, 255, 0.5); }

.taskTextOutside0,
.taskTextOutside2 {
  fill: lightgrey; }

.taskTextOutside1,
.taskTextOutside3 {
  fill: lightgrey; }

/* Active task */
.active0,
.active1,
.active2,
.active3 {
  fill: #81B1DB;
  stroke: rgba(255, 255, 255, 0.5); }

.activeText0,
.activeText1,
.activeText2,
.activeText3 {
  fill: #323D47 !important; }

/* Completed task */
.done0,
.done1,
.done2,
.done3 {
  stroke: grey;
  fill: lightgrey;
  stroke-width: 2; }

.doneText0,
.doneText1,
.doneText2,
.doneText3 {
  fill: #323D47 !important; }

/* Tasks on the critical line */
.crit0,
.crit1,
.crit2,
.crit3 {
  stroke: #E83737;
  fill: #E83737;
  stroke-width: 2; }

.activeCrit0,
.activeCrit1,
.activeCrit2,
.activeCrit3 {
  stroke: #E83737;
  fill: #81B1DB;
  stroke-width: 2; }

.doneCrit0,
.doneCrit1,
.doneCrit2,
.doneCrit3 {
  stroke: #E83737;
  fill: lightgrey;
  stroke-width: 2;
  cursor: pointer;
  shape-rendering: crispEdges; }

.milestone {
  transform: rotate(45deg) scale(0.8, 0.8); }

.milestoneText {
  font-style: italic; }

.doneCritText0,
.doneCritText1,
.doneCritText2,
.doneCritText3 {
  fill: #323D47 !important; }

.activeCritText0,
.activeCritText1,
.activeCritText2,
.activeCritText3 {
  fill: #323D47 !important; }

.titleText {
  text-anchor: middle;
  font-size: 18px;
  fill: #323D47;
   }

g.classGroup text {
  fill: #9370DB;
  stroke: none;
  
  font-size: 10px; }
  g.classGroup text .title {
    font-weight: bolder; }

g.classGroup rect {
  fill: #BDD5EA;
  stroke: #9370DB; }

g.classGroup line {
  stroke: #9370DB;
  stroke-width: 1; }

.classLabel .box {
  stroke: none;
  stroke-width: 0;
  fill: #BDD5EA;
  opacity: 0.5; }

.classLabel .label {
  fill: #9370DB;
  font-size: 10px; }

.relation {
  stroke: #9370DB;
  stroke-width: 1;
  fill: none; }

#compositionStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#compositionEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#aggregationStart {
  fill: #BDD5EA;
  stroke: #9370DB;
  stroke-width: 1; }

#aggregationEnd {
  fill: #BDD5EA;
  stroke: #9370DB;
  stroke-width: 1; }

#dependencyStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#dependencyEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#extensionStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#extensionEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

.commit-id,
.commit-msg,
.branch-label {
  fill: lightgrey;
  color: lightgrey;
   }

.pieTitleText {
  text-anchor: middle;
  font-size: 25px;
  fill: #eee;
}

g.stateGroup text {
  stroke: none;
  font-size: 10px;
}

g.stateGroup circle {
  fill: white !important;
  stroke: white !important;
}

g.stateGroup .state-title {
  font-weight: bolder;
  fill: black; }

g.stateGroup rect {
  fill: #ececff;
  stroke: #9370DB; }

g.stateGroup line {
  stroke: #9370DB;
  stroke-width: 1; }

.transition {
  stroke: #9370DB;
  stroke-width: 1;
  fill: none; }

.stateGroup .composit {
  fill: #555;
  border-bottom: 1px; }

.state-note {
  stroke: rgba(255, 255, 255, 0.25);
  fill: #fff5ad; }
  .state-note text {
    fill: black;
    stroke: none;
    font-size: 10px; }

.stateLabel .box {
  stroke: none;
  stroke-width: 0;
  fill: #BDD5EA;
  opacity: 0.5; }

.stateLabel text {
  fill: black;
  font-size: 10px;
  font-weight: bold;
}

.cluster-label {
  color:black;
}

.statediagram-cluster rect {
  fill: #BDD5EA;
  stroke: #9370DB; 
  stroke-width: 1px;
}
.statediagram-cluster rect.outer {
  rx: 5px;
  ry: 5px;
}
.statediagram-state .divider {
  stroke: #9370DB; 
}

.statediagram-state .title-state {
  rx: 5px;
  ry: 5px;
}
.statediagram-cluster.statediagram-cluster .inner {
  fill: white;
}
.statediagram-cluster.statediagram-cluster-alt .inner {
  fill: #e0e0e0;
}

.statediagram-cluster .inner {
  rx:0;
  ry:0;
}

.statediagram-state rect.basic {
  rx: 5px;
  ry: 5px;
}
.statediagram-state rect.divider {
  stroke-dasharray: 10,10;
  fill: #efefef;
}

.note-edge {
  stroke-dasharray: 5;
}

.statediagram-note rect {
  stroke: var(--cluster-border);
  fill: #fff5ad;
  stroke-width: 1px;
  rx: 0;
  ry: 0;
}

.node circle.state-start {
  fill: black;
  stroke: black;
}
.node circle.state-end {
  fill: black;
  stroke: white;
  stroke-width: 1.5
}
#statediagram-barbEnd {
  fill: #9370DB; 
}

/* CSS Document */

/** code highlight */

.cm-s-inner .cm-variable,
.cm-s-inner .cm-operator,
.cm-s-inner .cm-property {
    color: #b8bfc6;
}

.cm-s-inner .cm-keyword {
    color: #C88FD0;
}

.cm-s-inner .cm-tag {
    color: #7DF46A;
}

.cm-s-inner .cm-attribute {
    color: #7575E4;
}

.CodeMirror div.CodeMirror-cursor {
    border-left: 1px solid #b8bfc6;
    z-index: 3;
}

.cm-s-inner .cm-string {
    color: #D26B6B;
}

.cm-s-inner .cm-comment,
.cm-s-inner.cm-comment {
    color: #DA924A;
}

.cm-s-inner .cm-header,
.cm-s-inner .cm-def,
.cm-s-inner.cm-header,
.cm-s-inner.cm-def {
    color: #8d8df0;
}

.cm-s-inner .cm-quote,
.cm-s-inner.cm-quote {
    color: #57ac57;
}

.cm-s-inner .cm-hr {
    color: #d8d5d5;
}

.cm-s-inner .cm-link {
    color: #d3d3ef;
}

.cm-s-inner .cm-negative {
    color: #d95050;
}

.cm-s-inner .cm-positive {
    color: #50e650;
}

.cm-s-inner .cm-string-2 {
    color: #f50;
}

.cm-s-inner .cm-meta,
.cm-s-inner .cm-qualifier {
    color: #b7b3b3;
}

.cm-s-inner .cm-builtin {
    color: #f3b3f8;
}

.cm-s-inner .cm-bracket {
    color: #997;
}

.cm-s-inner .cm-atom,
.cm-s-inner.cm-atom {
    color: #84B6CB;
}

.cm-s-inner .cm-number {
    color: #64AB8F;
}

.cm-s-inner .cm-variable {
    color: #b8bfc6;
}

.cm-s-inner .cm-variable-2 {
    color: #9FBAD5;
}

.cm-s-inner .cm-variable-3 {
    color: #1cc685;
}

.CodeMirror-selectedtext,
.CodeMirror-selected {
    background: #4a89dc;
    color: #fff !important;
    text-shadow: none;
}

.CodeMirror-gutters {
    border-right: none;
}

/* CSS Document */

/** markdown source **/
.cm-s-typora-default .cm-header, 
.cm-s-typora-default .cm-property
{
    color: #cebcca;
}

.CodeMirror.cm-s-typora-default div.CodeMirror-cursor{
    border-left: 3px solid #b8bfc6;
}

.cm-s-typora-default .cm-comment {
    color: #9FB1FF;
}

.cm-s-typora-default .cm-string {
    color: #A7A7D9
}

.cm-s-typora-default .cm-atom, .cm-s-typora-default .cm-number {
    color: #848695;
    font-style: italic;
}

.cm-s-typora-default .cm-link {
    color: #95B94B;
}

.cm-s-typora-default .CodeMirror-activeline-background {
    background: rgba(51, 51, 51, 0.72);
}

.cm-s-typora-default .cm-comment, .cm-s-typora-default .cm-code {
	color: #8aa1e1;
}@import "";
@import "";
@import "";

:root {
    --bg-color:  #363B40;
    --side-bar-bg-color: #2E3033;
    --text-color: #b8bfc6;

    --select-text-bg-color:#4a89dc;

    --item-hover-bg-color: #0a0d16;
    --control-text-color: #b7b7b7;
    --control-text-hover-color: #eee;
    --window-border: 1px solid #555;

    --active-file-bg-color: rgb(34, 34, 34);
    --active-file-border-color: #8d8df0;

    --primary-color: #a3d5fe;

    --active-file-text-color: white;
    --item-hover-bg-color: #70717d;
    --item-hover-text-color: white;
    --primary-color: #6dc1e7;

    --rawblock-edit-panel-bd: #333;

    --search-select-bg-color: #428bca;
}

html {
    font-size: 16px;
}

html,
body {
    -webkit-text-size-adjust: 100%;
    -ms-text-size-adjust: 100%;
    background: #363B40;
    background: var(--bg-color);
    fill: currentColor;
    line-height: 1.625rem;
}

#write {
    max-width: 1080px;
}


@media only screen and (min-width: 1400px) {
	#write {
		max-width: 1024px;
	}
}

@media only screen and (min-width: 1800px) {
	#write {
		max-width: 1200px;
	}
}

html,
body,
button,
input,
select,
textarea,
div.code-tooltip-content {
    color: #b8bfc6;
    border-color: transparent;
}

div.code-tooltip,
.md-hover-tip .md-arrow:after {
    background: #333;
}

.popover.bottom > .arrow:after {
    border-bottom-color: #333;
}

html,
body,
button,
input,
select,
textarea {
    font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
}

hr {
    height: 2px;
    border: 0;
    margin: 24px 0 !important;
}

h1,
h2,
h3,
h4,
h5,
h6 {
    font-family: "Lucida Grande", "Corbel", sans-serif;
    font-weight: normal;
    clear: both;
    -ms-word-wrap: break-word;
    word-wrap: break-word;
    margin: 0;
    padding: 0;
    color: #DEDEDE
}

h1 {
    font-size: 2.5rem;
    /* 36px */
    line-height: 2.75rem;
    /* 40px */
    margin-bottom: 1.5rem;
    /* 24px */
    letter-spacing: -1.5px;
}

h2 {
    font-size: 1.63rem;
    /* 24px */
    line-height: 1.875rem;
    /* 30px */
    margin-bottom: 1.5rem;
    /* 24px */
    letter-spacing: -1px;
    font-weight: bold;
}

h3 {
    font-size: 1.17rem;
    /* 18px */
    line-height: 1.5rem;
    /* 24px */
    margin-bottom: 1.5rem;
    /* 24px */
    letter-spacing: -1px;
    font-weight: bold;
}

h4 {
    font-size: 1.12rem;
    /* 16px */
    line-height: 1.375rem;
    /* 22px */
    margin-bottom: 1.5rem;
    /* 24px */
    color: white;
}

h5 {
    font-size: 0.97rem;
    /* 16px */
    line-height: 1.25rem;
    /* 22px */
    margin-bottom: 1.5rem;
    /* 24px */
    font-weight: bold;
}

h6 {
    font-size: 0.93rem;
    /* 16px */
    line-height: 1rem;
    /* 16px */
    margin-bottom: 0.75rem;
    color: white;
}

@media (min-width: 980px) {
    h3.md-focus:before,
    h4.md-focus:before,
    h5.md-focus:before,
    h6.md-focus:before {
        color: #ddd;
        border: 1px solid #ddd;
        border-radius: 3px;
        position: absolute;
        left: -1.642857143rem;
        top: .357142857rem;
        float: left;
        font-size: 9px;
        padding-left: 2px;
        padding-right: 2px;
        vertical-align: bottom;
        font-weight: normal;
        line-height: normal;
    }

    h3.md-focus:before {
        content: 'h3';
    }

    h4.md-focus:before {
        content: 'h4';
    }

    h5.md-focus:before {
        content: 'h5';
        top: 0px;
    }

    h6.md-focus:before {
        content: 'h6';
        top: 0px;
    }
}

a {
    text-decoration: none;
    outline: 0;
}

a:hover {
    outline: 0;
}

a:focus {
    outline: thin dotted;
}

sup.md-footnote {
    background-color: #555;
    color: #ddd;
}

p {
    -ms-word-wrap: break-word;
    word-wrap: break-word;
}

p,
ul,
dd,
ol,
hr,
address,
pre,
table,
iframe,
.wp-caption,
.wp-audio-shortcode,
.wp-video-shortcode {
    margin-top: 0;
    margin-bottom: 1.5rem;
    /* 24px */
}

li > blockquote {
	margin-bottom: 0;
}

audio:not([controls]) {
    display: none;
}

[hidden] {
    display: none;
}

::-moz-selection {
    background: #4a89dc;
    color: #fff;
    text-shadow: none;
}

*.in-text-selection,
::selection {
    background: #4a89dc;
    color: #fff;
    text-shadow: none;
}

ul,
ol {
    padding: 0 0 0 1.875rem;
    /* 30px */
}

ul {
    list-style: square;
}

ol {
    list-style: decimal;
}

ul ul,
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<div id='write'  class=''><h2><a name="第三章有模型数值迭代" class="md-header-anchor"></a><span>第三章：有模型数值迭代</span></h2><p><span>对于实际问题，由于直接求解 Bellman 期望方程和 Bellman 最优方程往往有困难（例如需要极多的计算资源），本章在动力系统完全已知的情况下，用迭代的数值方法来求解。由于有模型迭代并没有从数据中学习，所以一般不认为是一种机器学习或强化学习方法。</span></p><h3><a name="一度量空间与压缩映射" class="md-header-anchor"></a><span>一、度量空间与压缩映射</span></h3><p><span>有模型策略迭代的理论基础：度量空间上的 Banach 不动点定理。下面简单介绍必要的概念，并证明 Bellman 算子是压缩映射，可以用 Banach 不动点定理迭代求解 Bellman 方程。</span></p><p><strong><span>度量</span></strong><span>（mertic，又称距离），是定义在集合上的二元函数。对于集合 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.874ex" height="1.842ex" viewBox="0 -749.6 807 793.1" role="img" focusable="false" style="vertical-align: -0.101ex;"><defs><path stroke-width="0" id="E296-MJCAL-58" d="M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 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xlink:href="#E252-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E252-MJMAIN-3A" x="800" y="0"></use><use xlink:href="#E252-MJCAL-58" x="1356" y="0"></use><use xlink:href="#E252-MJMAIN-D7" x="2385" y="0"></use><use xlink:href="#E252-MJCAL-58" x="3386" y="0"></use><use xlink:href="#E252-MJMAIN-2192" x="4470" y="0"></use><use xlink:href="#E252-MJAMS-52" x="5748" y="0"></use></g></svg></span><script type="math/tex">d: \mathcal X \times \mathcal X \rightarrow \Bbb R</script><span> ，需满足：</span></p><ul><li><p><span>非负性：对任意的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.223ex" height="2.517ex" viewBox="0 -832.7 4401.6 1083.8" role="img" focusable="false" style="vertical-align: -0.583ex;"><defs><path stroke-width="0" id="E275-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E275-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path stroke-width="0" id="E275-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E275-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="0" id="E275-MJCAL-58" d="M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 413 551 442T648 509T705 561Q707 565 707 578Q707 610 682 614Q667 614 667 626Q667 641 695 662T755 683Q765 683 775 680T796 662T807 623Q807 596 792 572T713 499T530 376L505 361V356Q508 346 511 278T524 148T557 75Q569 69 580 69Q585 69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E275-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032" x="808" y="513"></use><use xlink:href="#E275-MJMAIN-2C" x="866" y="0"></use><g transform="translate(1311,0)"><use xlink:href="#E275-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E275-MJMAIN-2208" x="2649" y="0"></use><use xlink:href="#E275-MJCAL-58" x="3594" y="0"></use></g></svg></span><script type="math/tex">x', x'' \in \mathcal X</script><span> ，有 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.177ex" height="2.71ex" viewBox="0 -832.7 5673.3 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E254-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 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280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E254-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path stroke-width="0" id="E254-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E254-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E254-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path stroke-width="0" id="E254-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E254-MJMATHI-64" x="0" y="0"></use><g transform="translate(689,0)"><use xlink:href="#E254-MJMAIN-28"></use><g transform="translate(389,0)"><use xlink:href="#E254-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E254-MJMAIN-2032" x="808" y="513"></use></g><use xlink:href="#E254-MJMAIN-2C" x="1255" y="0"></use><g transform="translate(1700,0)"><use xlink:href="#E254-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E254-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E254-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E254-MJMAIN-29" x="2761" y="0"></use></g><use xlink:href="#E254-MJMAIN-2265" x="4117" y="0"></use><use xlink:href="#E254-MJMAIN-30" x="5173" y="0"></use></g></svg></span><script type="math/tex">d\left(x', x''\right) \ge 0</script><span> ；</span></p></li><li><p><span>同一性：对任意的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.223ex" height="2.517ex" viewBox="0 -832.7 4401.6 1083.8" role="img" focusable="false" style="vertical-align: -0.583ex;"><defs><path stroke-width="0" id="E275-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E275-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path stroke-width="0" id="E275-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E275-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="0" id="E275-MJCAL-58" d="M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 413 551 442T648 509T705 561Q707 565 707 578Q707 610 682 614Q667 614 667 626Q667 641 695 662T755 683Q765 683 775 680T796 662T807 623Q807 596 792 572T713 499T530 376L505 361V356Q508 346 511 278T524 148T557 75Q569 69 580 69Q585 69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E275-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032" x="808" y="513"></use><use xlink:href="#E275-MJMAIN-2C" x="866" y="0"></use><g transform="translate(1311,0)"><use xlink:href="#E275-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E275-MJMAIN-2208" x="2649" y="0"></use><use xlink:href="#E275-MJCAL-58" x="3594" y="0"></use></g></svg></span><script type="math/tex">x', x'' \in \mathcal X</script><span> ，如果 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.177ex" height="2.71ex" viewBox="0 -832.7 5673.3 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E256-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 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transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E256-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E256-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E256-MJMAIN-29" x="2761" y="0"></use></g><use xlink:href="#E256-MJMAIN-3D" x="4117" y="0"></use><use xlink:href="#E256-MJMAIN-30" x="5173" y="0"></use></g></svg></span><script type="math/tex">d\left(x', x''\right) = 0</script><span> ，则 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.574ex" height="2.131ex" viewBox="0 -832.7 3260.9 917.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E257-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E257-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path stroke-width="0" id="E257-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E257-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E257-MJMAIN-2032" x="808" y="513"></use><use xlink:href="#E257-MJMAIN-3D" x="1144" y="0"></use><g transform="translate(2200,0)"><use xlink:href="#E257-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E257-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E257-MJMAIN-2032" x="275" y="0"></use></g></g></g></svg></span><script type="math/tex">x'=x''</script><span> ；</span></p></li><li><p><span>对称性：对任意的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.223ex" height="2.517ex" viewBox="0 -832.7 4401.6 1083.8" role="img" focusable="false" style="vertical-align: -0.583ex;"><defs><path stroke-width="0" id="E275-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E275-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path stroke-width="0" id="E275-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E275-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="0" id="E275-MJCAL-58" d="M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 413 551 442T648 509T705 561Q707 565 707 578Q707 610 682 614Q667 614 667 626Q667 641 695 662T755 683Q765 683 775 680T796 662T807 623Q807 596 792 572T713 499T530 376L505 361V356Q508 346 511 278T524 148T557 75Q569 69 580 69Q585 69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E275-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032" x="808" y="513"></use><use xlink:href="#E275-MJMAIN-2C" x="866" y="0"></use><g transform="translate(1311,0)"><use xlink:href="#E275-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E275-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E275-MJMAIN-2208" x="2649" y="0"></use><use xlink:href="#E275-MJCAL-58" x="3594" y="0"></use></g></svg></span><script type="math/tex">x', x'' \in \mathcal X</script><span> ，有 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="20.933ex" height="2.71ex" viewBox="0 -832.7 9012.9 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E259-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E259-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E259-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E259-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path stroke-width="0" id="E259-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E259-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E259-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E259-MJMATHI-64" x="0" y="0"></use><g transform="translate(689,0)"><use xlink:href="#E259-MJMAIN-28"></use><g transform="translate(389,0)"><use xlink:href="#E259-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E259-MJMAIN-2032" x="808" y="513"></use></g><use xlink:href="#E259-MJMAIN-2C" x="1255" y="0"></use><g transform="translate(1700,0)"><use xlink:href="#E259-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E259-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E259-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E259-MJMAIN-29" x="2761" y="0"></use></g><use xlink:href="#E259-MJMAIN-3D" x="4117" y="0"></use><use xlink:href="#E259-MJMATHI-64" x="5173" y="0"></use><g transform="translate(5862,0)"><use xlink:href="#E259-MJMAIN-28"></use><g transform="translate(389,0)"><use xlink:href="#E259-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E259-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E259-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E259-MJMAIN-2C" x="1449" y="0"></use><g transform="translate(1894,0)"><use xlink:href="#E259-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E259-MJMAIN-2032" x="808" y="513"></use></g><use xlink:href="#E259-MJMAIN-29" x="2761" y="0"></use></g></g></svg></span><script type="math/tex">d\left(x', x''\right) = d\left(x'',x'\right)</script><span> ；</span></p></li><li><p><span>三角不等式：对任意的 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.172ex" height="2.517ex" viewBox="0 -832.7 6101.6 1083.8" role="img" focusable="false" style="vertical-align: -0.583ex;"><defs><path stroke-width="0" id="E260-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E260-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path stroke-width="0" id="E260-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 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69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E260-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E260-MJMAIN-2032" x="808" y="513"></use><use xlink:href="#E260-MJMAIN-2C" x="866" y="0"></use><g transform="translate(1311,0)"><use xlink:href="#E260-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E260-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E260-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E260-MJMAIN-2C" x="2372" y="0"></use><g transform="translate(2816,0)"><use xlink:href="#E260-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E260-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E260-MJMAIN-2032" x="275" y="0"></use><use transform="scale(0.707)" xlink:href="#E260-MJMAIN-2032" x="550" y="0"></use></g></g><use xlink:href="#E260-MJMAIN-2208" x="4349" y="0"></use><use xlink:href="#E260-MJCAL-58" x="5294" y="0"></use></g></svg></span><script type="math/tex">x', x'', x''' \in \mathcal X</script><span> ，有 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="34.046ex" height="2.71ex" viewBox="0 -832.7 14658.5 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E261-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E261-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E261-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E261-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path stroke-width="0" id="E261-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E261-MJMATHI-64" x="0" y="0"></use><g transform="translate(689,0)"><use xlink:href="#E261-MJMAIN-28"></use><g transform="translate(389,0)"><use xlink:href="#E261-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032" x="808" y="513"></use></g><use xlink:href="#E261-MJMAIN-2C" x="1255" y="0"></use><g transform="translate(1700,0)"><use xlink:href="#E261-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032" x="275" y="0"></use><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032" x="550" y="0"></use></g></g><use xlink:href="#E261-MJMAIN-29" x="2955" y="0"></use></g><use xlink:href="#E261-MJMAIN-2264" x="4311" y="0"></use><use xlink:href="#E261-MJMATHI-64" x="5367" y="0"></use><g transform="translate(6057,0)"><use xlink:href="#E261-MJMAIN-28"></use><g transform="translate(389,0)"><use xlink:href="#E261-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032" x="808" y="513"></use></g><use xlink:href="#E261-MJMAIN-2C" x="1255" y="0"></use><g transform="translate(1700,0)"><use xlink:href="#E261-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E261-MJMAIN-29" x="2761" y="0"></use></g><use xlink:href="#E261-MJMAIN-2B" x="9429" y="0"></use><use xlink:href="#E261-MJMATHI-64" x="10429" y="0"></use><g transform="translate(11119,0)"><use xlink:href="#E261-MJMAIN-28"></use><g transform="translate(389,0)"><use xlink:href="#E261-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E261-MJMAIN-2C" x="1449" y="0"></use><g transform="translate(1894,0)"><use xlink:href="#E261-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032" x="275" y="0"></use><use transform="scale(0.707)" xlink:href="#E261-MJMAIN-2032" x="550" y="0"></use></g></g><use xlink:href="#E261-MJMAIN-29" x="3149" y="0"></use></g></g></svg></span><script type="math/tex">d\left(x', x'''\right) \le d\left(x', x''\right) + d\left(x'', x'''\right)</script><span> ；</span></p><p><em><span>度量看起来与范数很像，但其本质是不一样的，参考资料：</span><a href='https://www.zhihu.com/question/42312263' target='_blank' class='url'>https://www.zhihu.com/question/42312263</a></em></p></li></ul><p><span>有序对 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.929ex" height="2.71ex" viewBox="0 -832.7 2552.7 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E291-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E291-MJCAL-58" d="M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 413 551 442T648 509T705 561Q707 565 707 578Q707 610 682 614Q667 614 667 626Q667 641 695 662T755 683Q765 683 775 680T796 662T807 623Q807 596 792 572T713 499T530 376L505 361V356Q508 346 511 278T524 148T557 75Q569 69 580 69Q585 69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z"></path><path stroke-width="0" id="E291-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E291-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E291-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E291-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E291-MJCAL-58" x="389" y="0"></use><use xlink:href="#E291-MJMAIN-2C" x="1196" y="0"></use><use xlink:href="#E291-MJMATHI-64" x="1640" y="0"></use><use xlink:href="#E291-MJMAIN-29" x="2163" y="0"></use></g></svg></span><script type="math/tex">(\mathcal X, d)</script><span> 又称为</span><strong><span>度量空间</span></strong><span>（metric space）。考虑有限 Markov 决策过程状态函数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.895ex" height="2.71ex" viewBox="0 -832.7 5121.3 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E263-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 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263Q178 260 177 258Q172 247 164 239T151 227T136 218L127 213L124 202Q118 186 118 163Q120 124 165 86T292 48Q374 48 423 86T473 186V193Q473 267 347 327Q268 364 239 389Q191 431 191 486Q191 547 242 600T356 679T470 705Q472 705 478 705T489 704Q551 704 596 682T642 610Q642 566 621 545Q592 516 554 512Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E263-MJMATHI-76" x="0" y="0"></use><use xlink:href="#E263-MJMAIN-28" x="485" y="0"></use><use xlink:href="#E263-MJMATHI-73" x="874" y="0"></use><use xlink:href="#E263-MJMAIN-29" x="1343" y="0"></use><use xlink:href="#E263-MJMAIN-28" x="2009" y="0"></use><use xlink:href="#E263-MJMATHI-73" x="2398" y="0"></use><use xlink:href="#E263-MJMAIN-2208" x="3145" y="0"></use><use xlink:href="#E263-MJCAL-53" x="4090" y="0"></use><use xlink:href="#E263-MJMAIN-29" x="4732" y="0"></use></g></svg></span><script type="math/tex">v(s) \; (s \in \mathcal S)</script><span> ，其中所有可能的取值组成集合 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x="935" y="0"></use><g transform="translate(1991,0)"><use xlink:href="#E264-MJAMS-52" x="0" y="0"></use><g transform="translate(722,409)"><use transform="scale(0.707)" xlink:href="#E264-MJMAIN-7C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E264-MJCAL-53" x="278" y="0"></use><use transform="scale(0.707)" xlink:href="#E264-MJMAIN-7C" x="920" y="0"></use></g></g></g></svg></span><script type="math/tex">\mathcal V = \Bbb R^{|\mathcal S|}</script><span> ，定义 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.082ex" height="2.228ex" viewBox="0 -749.6 1327.1 959.2" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E266-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E266-MJMAIN-221E" d="M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 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id="MathJax-Element-234">d_\infty \left(v',v''\right)=\max_{s \in \mathcal S} \; \big|v'\left(s\right)-v''\left(s\right)\big|</script></div></div><p><span>可以证明 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.082ex" height="2.228ex" viewBox="0 -749.6 1327.1 959.2" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E266-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 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type="math/tex">d_\infty</script><span> 是 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.528ex" height="2.035ex" viewBox="0 -749.6 658 876.2" role="img" focusable="false" style="vertical-align: -0.294ex;"><defs><path stroke-width="0" id="E267-MJCAL-56" d="M25 633Q25 647 47 665T100 683Q291 683 291 306Q291 264 288 213T282 132L279 102Q281 102 308 126T378 191T464 279T545 381T596 479Q600 490 600 502Q600 527 581 550T523 577Q505 577 505 601Q505 622 516 647T542 681Q546 683 558 683Q605 679 631 645T658 559Q658 423 487 215Q409 126 308 37T190 -52Q177 -52 177 -28Q177 -26 183 15T196 127T203 270Q203 356 192 421T165 523T126 583T83 613T41 620Q25 620 25 633Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E267-MJCAL-56" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathcal V</script><span> 上的一个度量。证明：显然满足非负性、同一性、对称性，而对于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.711ex" height="1.939ex" viewBox="0 -749.6 2889.6 834.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E268-MJMAIN-2200" d="M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z"></path><path stroke-width="0" id="E268-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 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y="0"></use><use xlink:href="#E237-MJMATHI-73" x="389" y="0"></use><use xlink:href="#E237-MJMAIN-29" x="858" y="0"></use></g><g transform="translate(19662,0)"><g transform="translate(0,850)"><use xlink:href="#E237-MJMAIN-2223" x="0" y="-750"></use><use xlink:href="#E237-MJMAIN-2223" x="0" y="-951"></use></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-235">\begin{split}
\big|v'\left(s\right)-v'''\left(s\right)\big| 
&= \big|\;[v'\left(s\right)-v''\left(s\right)] + [v''\left(s\right)-v'''\left(s\right)]\;\big| \\
&\le \big|\;v'\left(s\right) - v''\left(s\right)\;\big| + \big|\;v''\left(s\right)-v'''\left(s\right)\;\big| \\
&\le \max_{s \in \mathcal S} \; \big|v'\left(s\right) - v''\left(s\right)\big| + \max_{s \in \mathcal S} \; \big|v''\left(s\right) - v'''\left(s\right)\big|
\end{split}</script></div></div><p><span>可得三角不等式，所以 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.45ex" height="2.71ex" viewBox="0 -832.7 3207.8 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E280-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E280-MJCAL-56" d="M25 633Q25 647 47 665T100 683Q291 683 291 306Q291 264 288 213T282 132L279 102Q281 102 308 126T378 191T464 279T545 381T596 479Q600 490 600 502Q600 527 581 550T523 577Q505 577 505 601Q505 622 516 647T542 681Q546 683 558 683Q605 679 631 645T658 559Q658 423 487 215Q409 126 308 37T190 -52Q177 -52 177 -28Q177 -26 183 15T196 127T203 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transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E280-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E280-MJCAL-56" x="389" y="0"></use><use xlink:href="#E280-MJMAIN-2C" x="1047" y="0"></use><g transform="translate(1491,0)"><use xlink:href="#E280-MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E280-MJMAIN-221E" x="735" y="-213"></use></g><use xlink:href="#E280-MJMAIN-29" x="2818" y="0"></use></g></svg></span><script type="math/tex">(\mathcal V, d_\infty)</script><span> 是一个度量空间。</span></p><p><span>对于一个度量空间，如果 Cauchy 序列都收敛在该空间内，则称这个度量空间是</span><strong><span>完备的</span></strong><span>（complete）。例如，实数集 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.677ex" height="1.939ex" viewBox="0 -749.6 722 834.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E270-MJAMS-52" d="M17 665Q17 672 28 683H221Q415 681 439 677Q461 673 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E270-MJAMS-52" x="0" y="0"></use></g></svg></span><script type="math/tex">\Bbb R</script><span> 就是一个著名的完备空间（事实上实数集就是由完备性定义出来的。有理数集不完备，加上无理数集就完备了），对于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.45ex" height="2.71ex" viewBox="0 -832.7 3207.8 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E280-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E280-MJCAL-56" d="M25 633Q25 647 47 665T100 683Q291 683 291 306Q291 264 288 213T282 132L279 102Q281 102 308 126T378 191T464 279T545 381T596 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369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E280-MJMAIN-221E" d="M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z"></path><path stroke-width="0" id="E280-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 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xlink:href="#E276-MJMAIN-28"></use><g transform="translate(389,0)"><use xlink:href="#E276-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E276-MJMAIN-2032" x="808" y="513"></use></g><use xlink:href="#E276-MJMAIN-2C" x="1255" y="0"></use><g transform="translate(1700,0)"><use xlink:href="#E276-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E276-MJMAIN-2032"></use><use transform="scale(0.707)" xlink:href="#E276-MJMAIN-2032" x="275" y="0"></use></g></g><use xlink:href="#E276-MJMAIN-29" x="2761" y="0"></use></g></g></svg></span><script type="math/tex">d\left(\, t(x'), t(x'')\,\right) < \gamma\, d\left(x',x''\right)</script><span> ，则称映射 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.838ex" height="1.842ex" viewBox="0 -708.1 361 793.1" role="img" focusable="false" style="vertical-align: 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\label{eq:2}</script></div></div><ul><li><span>Bellman 最优算子 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.497ex" height="2.228ex" viewBox="0 -749.6 4519.7 959.2" role="img" focusable="false" style="vertical-align: -0.291ex; margin-bottom: -0.196ex;"><defs><path stroke-width="0" id="E284-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E284-MJMAIN-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 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288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E289-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script><span> 是映射 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.838ex" height="1.842ex" viewBox="0 -708.1 361 793.1" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E293-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E293-MJMATHI-74" x="0" y="0"></use></g></svg></span><script type="math/tex">t</script><span> 的</span><strong><span>不动点</span></strong><span>（fix point）。在完备度量空间上的压缩映射有非常重要的结论：</span><strong><span>Banach 不动点定理</span></strong><span>（Banach fixed-point theorem，又称压缩映射定理，compressed mapping theorem）。其内容为： </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.929ex" height="2.71ex" viewBox="0 -832.7 2552.7 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E291-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E291-MJCAL-58" d="M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 413 551 442T648 509T705 561Q707 565 707 578Q707 610 682 614Q667 614 667 626Q667 641 695 662T755 683Q765 683 775 680T796 662T807 623Q807 596 792 572T713 499T530 376L505 361V356Q508 346 511 278T524 148T557 75Q569 69 580 69Q585 69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z"></path><path stroke-width="0" id="E291-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E291-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E291-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E291-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E291-MJCAL-58" x="389" y="0"></use><use xlink:href="#E291-MJMAIN-2C" x="1196" y="0"></use><use xlink:href="#E291-MJMATHI-64" x="1640" y="0"></use><use xlink:href="#E291-MJMAIN-29" x="2163" y="0"></use></g></svg></span><script type="math/tex">(\mathcal X, d)</script><span> 是非空的完备度量空间， </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.136ex" height="1.939ex" viewBox="0 -749.6 4364.1 834.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E292-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 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191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z"></path><path stroke-width="0" id="E292-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E292-MJMATHI-74" x="0" y="0"></use><use xlink:href="#E292-MJMAIN-3A" x="638" y="0"></use><use xlink:href="#E292-MJCAL-58" x="1194" y="0"></use><use xlink:href="#E292-MJMAIN-2192" x="2279" y="0"></use><use xlink:href="#E292-MJCAL-58" x="3557" y="0"></use></g></svg></span><script type="math/tex">t:\mathcal X \rightarrow \mathcal X</script><span> 是一个压缩映射，则映射 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.838ex" height="1.842ex" viewBox="0 -708.1 361 793.1" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E293-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E293-MJMATHI-74" x="0" y="0"></use></g></svg></span><script type="math/tex">t</script><span> 在 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.874ex" height="1.842ex" viewBox="0 -749.6 807 793.1" role="img" focusable="false" style="vertical-align: -0.101ex;"><defs><path stroke-width="0" id="E296-MJCAL-58" d="M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 413 551 442T648 509T705 561Q707 565 707 578Q707 610 682 614Q667 614 667 626Q667 641 695 662T755 683Q765 683 775 680T796 662T807 623Q807 596 792 572T713 499T530 376L505 361V356Q508 346 511 278T524 148T557 75Q569 69 580 69Q585 69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E296-MJCAL-58" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathcal X</script><span> 内有且仅有一个不动点 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.481ex" height="1.842ex" viewBox="0 -500.4 1929.2 793.1" role="img" focusable="false" style="vertical-align: -0.68ex;"><defs><path stroke-width="0" id="E299-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E299-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E299-MJMAIN-221E" d="M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E299-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,-150)"><use transform="scale(0.707)" xlink:href="#E299-MJMAIN-2B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E299-MJMAIN-221E" x="778" y="0"></use></g></g></svg></span><script type="math/tex">x_{+\infty}</script><span> 。</span></p><p><span>不动点的求取方法：从 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.874ex" height="1.842ex" viewBox="0 -749.6 807 793.1" role="img" focusable="false" style="vertical-align: -0.101ex;"><defs><path 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xlink:href="#E296-MJCAL-58" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathcal X</script><span> 内的任意一个元素 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.382ex" height="1.649ex" viewBox="0 -500.4 1025.6 710.1" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E297-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E297-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E297-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E297-MJMAIN-30" x="808" y="-213"></use></g></svg></span><script type="math/tex">x_0</script><span> 开始，定义迭代序列 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="29.115ex" height="2.71ex" viewBox="0 -832.7 12535.4 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xlink:href="#E240-MJMAIN-31" x="808" y="-213"></use></g><use xlink:href="#E240-MJMAIN-2C" x="1414" y="0"></use><g transform="translate(1859,0)"><use xlink:href="#E240-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E240-MJMAIN-30" x="808" y="-213"></use></g><use xlink:href="#E240-MJMAIN-29" x="2884" y="0"></use></g></g><g transform="translate(0,-7341)"><use xlink:href="#E240-MJMAIN-3D" x="277" y="0"></use><g transform="translate(1333,0)"><g transform="translate(499,0)"><use xlink:href="#E240-MJMAIN-6C"></use><use xlink:href="#E240-MJMAIN-69" x="278" y="0"></use><use xlink:href="#E240-MJMAIN-6D" x="556" y="0"></use></g><g transform="translate(0,-600)"><use transform="scale(0.707)" xlink:href="#E240-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E240-MJMAIN-2192" x="600" y="0"></use><use transform="scale(0.707)" xlink:href="#E240-MJMAIN-2B" x="1600" y="0"></use><use transform="scale(0.707)" xlink:href="#E240-MJMAIN-221E" x="2378" y="0"></use></g></g><g transform="translate(3722,0)"><g transform="translate(286,0)"><rect stroke="none" width="5089" height="60" x="0" y="220"></rect><g transform="translate(60,693)"><use xlink:href="#E240-MJMATHI-3B3" x="0" y="0"></use><g transform="translate(550,362)"><use transform="scale(0.707)" xlink:href="#E240-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E240-MJMAIN-2032" x="736" y="595"></use></g><g transform="translate(1393,0)"><use xlink:href="#E240-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E240-MJMAIN-31" x="389" y="0"></use><use xlink:href="#E240-MJMAIN-2212" x="1111" y="0"></use><g transform="translate(2111,0)"><use xlink:href="#E240-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E240-MJMATHI-6E" x="778" y="513"></use></g><use xlink:href="#E240-MJMAIN-29" x="3186" y="0"></use></g></g><g transform="translate(1411,-686)"><use xlink:href="#E240-MJMAIN-31" x="0" y="0"></use><use xlink:href="#E240-MJMAIN-2212" x="722" y="0"></use><use xlink:href="#E240-MJMATHI-3B3" x="1722" y="0"></use></g></g></g><use xlink:href="#E240-MJMATHI-64" x="9217" y="0"></use><g transform="translate(9907,0)"><use xlink:href="#E240-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xlink:href="#E240-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E240-MJMAIN-31" x="808" y="-213"></use></g><use xlink:href="#E240-MJMAIN-2C" x="1414" y="0"></use><g transform="translate(1859,0)"><use xlink:href="#E240-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E240-MJMAIN-30" x="808" y="-213"></use></g><use xlink:href="#E240-MJMAIN-29" x="2884" y="0"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-238">\begin{split}
d\left(x_{k'},x_{k''}\right) &\le d\left(x_{k'},x_{k'+1}\right) + d\left(x_{k'+1},x_{k'+2}\right) + \cdots + d\left(x_{k''-1},x_{k''}\right) \\
&\le \sum_{k=k'}^{+\infty} d\left(x_{k+1},x_k\right) \\
&= \sum_{k=k'}^{+\infty} d\left(t\left(x_k\right),t\left(x_{k-1}\right)\right) \\
\text{（利用压缩映射）} &\le \sum_{k=k'}^{+\infty} \gamma d\left(x_{k},x_{k-1}\right) \\
\text{（反复利用压缩映射）} &\le \sum_{k=k'}^{+\infty} \gamma^k d\left(x_1,x_0\right) \\
&= \lim_{n \rightarrow +\infty} \frac{\gamma^{k'} \left(1-\gamma^n\right)}{1-\gamma} d\left(x_1,x_0\right)
\end{split}</script></div></div><p><span>由于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.263ex" height="2.71ex" viewBox="0 -832.7 3988.2 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E302-MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path><path stroke-width="0" id="E302-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 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y="0"></use><use xlink:href="#E302-MJMAIN-28" x="1765" y="0"></use><use xlink:href="#E302-MJMAIN-30" x="2154" y="0"></use><use xlink:href="#E302-MJMAIN-2C" x="2654" y="0"></use><use xlink:href="#E302-MJMAIN-31" x="3099" y="0"></use><use xlink:href="#E302-MJMAIN-29" x="3599" y="0"></use></g></svg></span><script type="math/tex">\gamma \in (0,1)</script><span> ，所以上式不等式右端可以任意小，证毕。另外，在证明过程中可以看出迭代正比于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.367ex" height="2.807ex" viewBox="0 -915.7 1019 1208.4" role="img" focusable="false" style="vertical-align: -0.68ex;"><defs><path stroke-width="0" id="E303-MJMATHI-3B3" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 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647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E303-MJMATHI-3B3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E303-MJMATHI-6B" x="778" y="513"></use></g></svg></span><script type="math/tex">\gamma^k</script><span> 的速度收敛。</span></p><p><span>有了不动点的求取方法，那么就可以用该方法求 Bellman 期望算子和最优算子的不动点。其中 Bellman 期望算子的不动点是策略价值，对应于策略迭代算法；Bellman 最优算子的不动点是最有价值，对应于价值迭代算法。</span></p><h3><a name="二有模型策略迭代" class="md-header-anchor"></a><span>二、有模型策略迭代</span></h3><p><span>各种策略迭代算法中包含以下的三个概念：</span></p><ul><li><strong><span>策略评估（policy evaluation）：</span></strong><span>对于给定的策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.331ex" height="1.36ex" viewBox="0 -500.4 573 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E320-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E320-MJMATHI-3C0" x="0" y="0"></use></g></svg></span><script type="math/tex">\pi</script><span> ，估计策略的价值，包括动作价值和状态价值。</span></li><li><strong><span>策略改进（policy improvement）：</span></strong><span>对于给定的策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.331ex" height="1.36ex" viewBox="0 -500.4 573 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E320-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E320-MJMATHI-3C0" x="0" y="0"></use></g></svg></span><script type="math/tex">\pi</script><span> ，在已知其价值函数的情况下，找到一个更优的策略。</span></li><li><strong><span>策略迭代（policy iteration）：</span></strong><span>综合利用策略评估和策略改进，找到最优策略。</span></li></ul><p><span>对于策略评估，由于状态价值函数有 </span><span class="MathJax_SVG" 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type="math/tex">|\mathcal S| \times |\mathcal A|</script><span> 个自变量，因此选择使用状态价值函数作为迭代存储变量，以节约空间。策略评估在迭代中使用 Bellman 期望方程的表达式更新一轮所有状态的状态价值函数，这样对所有状态价值函数的一次更新又称为一次扫描。策略评估迭代算法有以下两种：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n46" cid="n46" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-239-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; zoom: 0.971345;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="101.294ex" height="53.739ex" viewBox="-18.1 -43.5 43612.5 23137.6" role="img" focusable="false" style="vertical-align: -53.638ex; margin-left: -0.042ex; max-width: 100%;"><defs><path stroke-width="0" id="E241-MJMAINB-33" d="M80 503Q80 565 133 610T274 655Q366 655 421 623T491 538Q493 528 493 510Q493 446 453 407T361 348L376 344Q452 324 489 281T526 184Q526 152 514 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\large \textbf{算法 3-1   有模型策略评估迭代算法} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：动力系统 $p$ ，策略 $\pi$ 。} \\
&\text{输出：状态价值函数 $v_\pi$ 的估计值。} \\
&\text{参数：控制迭代次数的参数（例如容忍度误差 $\delta_{max}$ ，或最大迭代次数 $k_{max}$）。} \\
&\text{1.（初始化）对于 $s \in \mathcal S$ ，将 $v_0\left(s\right)$ 初始化为任意值（比如 0 ）。如果有终止状态，将终止状态初始化} \\
&\qquad \text{为 0 ，即 $v_0\left(s_{\text{终止}}\right) \leftarrow 0$ 。} \\
&\text{2.（迭代）对于 $k \leftarrow 0, 1, 2, 3, \cdots,$ 迭代执行以下步骤：} \\
&\qquad \text{2.1  对于 $s \in \mathcal S$ ，逐一更新 $v_{k+1}\left(s\right) \leftarrow \sum_a \pi \left(a \mid s \right) q_k\left(s,a\right)$ ，其中} \\
\end{split} \\

q_k\left(s,a\right) \leftarrow r\left(s,a\right) + \gamma \sum_{s'} p\left(s' \mid s, a\right) v_k\left(s'\right) \\

\begin{split}
&\qquad \qquad \;\, \text{2.2  若满足迭代终止条件（如对 $s \in \mathcal S$ 均有 $|v_{k+1}\left(s\right) - v_k\left(s\right)| \lt \delta_{max}$ ，或达到最大迭代次数} \\ 
&\qquad \qquad \qquad \; \text{$k=k_{max}$），则跳出循环。} \\
&\rule[-5pt]{275mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>事实上算法 3-1 没必要为每次扫描都重新分配一套空间 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.654ex" height="2.71ex" viewBox="0 -832.7 8031.7 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E308-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path stroke-width="0" id="E308-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 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\large \textbf{算法 3-2   有模型策略评估迭代算法（节省空间做法）} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：动力系统 $p$ ，策略 $\pi$ 。} \\
&\text{输出：$v_\pi$ 的估计值。} \\
&\text{参数：控制迭代次数的参数（例如容忍度误差 $\delta_{max}$ ，或最大迭代次数 $k_{max}$）。} \\
&\text{1.（初始化）$v_0\left(s\right) \leftarrow$ 任意值 $s \in \mathcal S$。如果有终止状态，将终止状态初始化为 0 ，即 $v_0\left(s_{\text{终止}}\right) \leftarrow 0$ 。} \\
&\text{2.（迭代）对于 $k \leftarrow 0, 1, 2, 3, \cdots,$ 迭代执行以下步骤：} \\
&\qquad \text{2.1  对于使用误差容忍度的情况，初始化本次迭代观测到的最大误差 } \delta \leftarrow 0\ ; \\
&\qquad \text{2.2  对于 $s \in \mathcal S$ ：} \\ 
&\qquad \qquad \text{2.2.1  计算新状态价值 } v_{\text{新}} \leftarrow \sum_a \pi \left(a \mid s \right) \left[r\left(s,a\right) + \gamma \sum_{s'} p\left(s' \mid s, a\right) v\left(s'\right)\right]\ ; \\
&\qquad \qquad \text{2.2.2  对于使用误差容忍度的情况，更新本次迭代观测到的最大误差 } \delta \leftarrow \max \left(\delta,\ |v_{\text{新}} - v\left(s\right)|\right)\ ; \\
&\qquad \qquad \text{2.2.3  更新状态价值函数 } v\left(s\right) \leftarrow v_{\text{新}}\ ; \\
&\qquad \text{2.3  若满足迭代终止条件（如 $\delta \lt \delta_{max}$ 或达 $k=k_{max}$），则跳出循环。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em} \\
\end{split}
\; \\ \; \\</script></div></div><p><span>算法 3-2 则只使用了一套存储空间，每次扫描时，他都及时更新状态价值函数。这样在更新后续的状态时，用来更新的状态价值函数有些在本次迭代中已经更新，有些在本次迭代中还没更新，所以，算法 3-2 的计算结果和算法 3-1 的计算结果不完全相同。不过，算法 3-2 在迭代次数不限的情况下也能收敛到状态价值函数。</span></p><p><span>对于策略改进，有</span><strong><span>策略改进定理：</span></strong><span>对于两个确定性的策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.331ex" height="1.36ex" viewBox="0 -500.4 573 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E320-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 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\label{eq:5}</script></div></div><p><span>在此基础上，若存在状态使得 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.968ex" height="2.71ex" viewBox="0 -832.7 1278 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E314-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E314-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="0" id="E314-MJMAIN-29" d="M60 749L64 750Q69 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xlink:href="#E316-MJMAIN-29" x="858" y="0"></use></g></g></svg></span><script type="math/tex">q_\pi\left(s,a\right) \gt v_\pi\left(s\right)</script><span> ，那么就可以构造一个新策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.017ex" height="2.131ex" viewBox="0 -832.7 868.4 917.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E317-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" 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\large \textbf{算法 3-3   有模型策略改进算法} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：动力系统 $p$ ，策略 $\pi$ 及其状态价值函数 $v_\pi$。} \\
&\text{输出：改进的策略 $\pi'$ ，或策略 $\pi$ 已经达到最优的标志。} \\
&\text{1. 对于每个状态 $s \in \mathcal S$ ，执行以下步骤：} \\
&\qquad \text{1.1  为每个动作 $a \in \mathcal A$ ，求得动作价值函数 } q_\pi \left(s,a\right) \leftarrow r\left(s,a\right) + \gamma \sum_{s'} p\left(s' \mid s, a\right) v_\pi \left(s'\right)\ ;\\
&\qquad \text{1.2  找到使得 $q_\pi \left(s,a\right)$ 最大的动作 $a$ ，即 $\pi'\left(s\right)=\underset{a}{\arg\max}\; q\left(s,a\right)$ 。} \\
&\text{2. 对于新策略 $\pi'$ 和旧策略 $\pi$ 相同，则说明旧策略已是最优；否则，输出改进的新策略 $\pi'$。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em} \\
\end{split}
\; \\ \; \\</script></div></div><p><span>算法 3-3 中，如果在后续不需要使用旧策略的情况下，可以不为新策略分配空间，由此就得到了算法 3-4 的版本。</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n58" cid="n58" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-244-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; zoom: 0.971345;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="101.294ex" height="42.935ex" viewBox="-18.1 -43.5 43612.5 18486" role="img" focusable="false" style="vertical-align: -42.834ex; margin-left: -0.042ex; max-width: 100%;"><defs><path stroke-width="0" id="E246-MJMAINB-33" d="M80 503Q80 565 133 610T274 655Q366 655 421 623T491 538Q493 528 493 510Q493 446 453 407T361 348L376 344Q452 324 489 281T526 184Q526 152 514 121T474 58T392 8T265 -11Q175 -11 111 34T48 152Q50 187 72 209T132 232Q171 232 193 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\large \textbf{算法 3-4   有模型策略改进算法（节约空间的做法）} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：动力系统 $p$ ，策略 $\pi$ 及其状态价值函数 $v$。} \\
&\text{输出：改进的策略（任然存储为 $\pi$ ），或策略已经达到最优的标志 $o$。} \\
&\text{1. 初始化原策略是否为最优的标记 $o$ 为 True。} \\
&\text{2. 对于每个状态 $s \in \mathcal S$ ，执行以下步骤：} \\
&\qquad \text{2.1  为每个动作 $a \in \mathcal A$ ，求得动作价值函数 } q \left(s,a\right) \leftarrow r\left(s,a\right) + \gamma \sum_{s'} p\left(s' \mid s, a\right) v \left(s'\right)\ ;\\
&\qquad \text{2.2  找到使得 $q \left(s,a\right)$ 最大的动作 $a'$ ，即 $a'=\underset{a}{\arg\max}\; q\left(s,a\right)$ ;} \\
&\qquad \text{2.3  如果 $\pi \left(s\right) \neq a'$ ，则更新 $\pi \left(s\right) \leftarrow a'$ ， $o \leftarrow$ False 。}\\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em} \\
\end{split}
\; \\ \; \\</script></div></div><p><span>在了解策略评估和策略改进后，通过策略迭代综合利用两者来获得最优策略。策略迭代从一个任意的确定性策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.377ex" height="1.649ex" viewBox="0 -500.4 1023.6 710.1" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E321-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E321-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E321-MJMATHI-3C0" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E321-MJMAIN-30" x="806" y="-213"></use></g></svg></span><script type="math/tex">\pi_0</script><span> 开始，交替进行策略评估和策略改进，其基本步骤顺序如下图所示：</span></p><div class="md-diagram-panel"><svg id="mermaidChart0" width="100%" xmlns="http://www.w3.org/2000/svg" style="max-width: 931.140625px;" viewBox="0 0 931.140625 54"><style>
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\large \textbf{算法 3-5   有模型策略迭代} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：动力系统 $p$ 。} \\
&\text{输出：最优策略。} \\
&\text{1.（初始化）将策略 $\pi_0$ 初始化为一个任意的确定性策略。} \\
&\text{2.（迭代）对于 $k \leftarrow 0, 1, 2, 3, \cdots$ ，执行以下步骤：} \\
&\qquad \text{2.1 （策略评估）使用策略评估算法，计算策略 $\pi_k$ 的状态价值函数 $v_{\pi_k}$ ;} \\
&\qquad \text{2.2 （策略更新）利用状态价值函数 $v_{\pi_k}$ 改进策略 $\pi_k$ ，得到改进的确定性策略 $\pi_{k+1}$ 。}  \\
&\qquad \qquad \text{如果 $\pi_k = \pi_{k+1}$（即对任意的 $s \in \mathcal S$ 均有 $\pi_k \left(s\right) = \pi_{k+1} \left(s\right)$ ），则迭代完成，} \\
&\qquad \qquad \text{返回策略 $\pi_k$ 为最终的最优策略。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em} \\
\end{split}
\; \\ \; \\</script></div></div><p><span>同样为了节省空间，在各次迭代中用相同的空间 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.075ex" height="2.71ex" viewBox="0 -832.7 6060.2 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E322-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path stroke-width="0" id="E322-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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stroke="none" width="41597" height="100" x="0" y="-500"></rect></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-246">\; \\ \; \\
\large \textbf{算法 3-6   有模型策略迭代（节约空间的版本）} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：动力系统 $p$ 。} \\
&\text{输出：最优策略 $\pi$ 。} \\
&\text{参数：策略评估需要的参数。} \\
&\text{1.（初始化）将策略 $\pi$ 初始化为一个任意的确定性策略。} \\
&\text{2.（迭代）迭代执行以下步骤：} \\
&\qquad \text{2.1 （策略评估）使用策略评估算法，计算策略 $\pi$ 的状态价值函数，存在 $v$ 中;} \\
&\qquad \text{2.2 （策略更新）利用 $v$ 中存储的状态价值函数进行策略改进，将改进的策略存储在 $\pi$ 。如果本次} \\
&\qquad \qquad \text{策略改进算法指示当前策略 $\pi$ 已经是最优策略，则迭代完成，返回策略 $\pi$ 为最终的最优策略。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em} \\
\end{split}
\; \\ \; \\</script></div></div><h3><a name="三有模型价值迭代" class="md-header-anchor"></a><span>三、有模型价值迭代</span></h3><p><span>策略评估迭代算法利用 Bellman 期望方程迭代求解给定策略的价值函数，而价值迭代算法则是利用 Bellman 最优方程迭代求解最优策略的价值函数，并进而求得最优策略，其算法如下：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n67" cid="n67" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-247-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; zoom: 0.971345;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="101.294ex" height="58.369ex" viewBox="-18.1 -43.5 43612.5 25131.2" role="img" focusable="false" style="vertical-align: -58.268ex; margin-left: -0.042ex; max-width: 100%;"><defs><path stroke-width="0" id="E249-MJMAINB-33" d="M80 503Q80 565 133 610T274 655Q366 655 421 623T491 538Q493 528 493 510Q493 446 453 407T361 348L376 344Q452 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\large \textbf{算法 3-7   有模型价值迭代算法} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：动力系统 $p$ 。} \\
&\text{输出：最优策略估计 $\pi$ 。} \\
&\text{参数：策略评估需要的参数。} \\
&\text{1.（初始化）$v_0\left(s\right) \leftarrow$ 任意值，$s \in \mathcal S$。如果有终止状态，$v_0\left(s_{\text{终止}}\right) \leftarrow 0$ 。} \\
&\text{2.（迭代）对于 $k \leftarrow 0, 1, 2, 3, \cdots,$ 迭代执行以下步骤：} \\
&\qquad \text{2.1  对于 $s \in \mathcal S$ ，逐一更新 } v_{k+1}\left(s\right) \leftarrow \max_{a} \left(r\left(s,a\right) + \gamma \sum_{s'} p\left(s' \mid s, a\right) v_k\left(s'\right)\right) ; \\
&\qquad \text{2.2  若满足容忍度误差（即对 $s \in \mathcal S$ 均有 $|v_{k+1}\left(s\right) - v_k\left(s\right)| \lt \delta_{max}$ ），或达到最大迭代} \\
&\qquad \qquad \text{次数（即 $k=k_{max}$），则跳出循环。} \\
&\text{3.（策略）根据价值函数输出确定性策略 $\pi_*$ ，使得：} \\
\end{split} \\
\pi_* \left(s\right) \leftarrow \underset{a}{\arg\max} \left(r\left(s,a\right) + \gamma \sum_{s'} p\left(s' \mid s,a\right) v_{k+1}\left(s'\right)\right); \qquad s \in \mathcal S \\
\begin{split}
&\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em} \\
\end{split}
\; \\ \; \\</script></div></div><p><span>同样，价值迭代也可以在存储状态价值函数时重复使用空间，因此有以下版本的价值迭代算法：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n69" cid="n69" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-248-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; zoom: 0.971345;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="101.294ex" height="67.63ex" viewBox="-18.1 -43.5 43612.5 29118.3" role="img" focusable="false" style="vertical-align: -67.529ex; margin-left: -0.042ex; max-width: 100%;"><defs><path stroke-width="0" id="E250-MJMAINB-33" d="M80 503Q80 565 133 610T274 655Q366 655 421 623T491 538Q493 528 493 510Q493 446 453 407T361 348L376 344Q452 324 489 281T526 184Q526 152 514 121T474 58T392 8T265 -11Q175 -11 111 34T48 152Q50 187 72 209T132 232Q171 232 193 208T216 147Q216 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\large \textbf{算法 3-8   有模型价值迭代（节约空间版本）} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：动力系统 $p$ 。} \\
&\text{输出：最优策略。} \\
&\text{参数：策略评估需要的参数。} \\
&\text{1.（初始化）$v_0\left(s\right) \leftarrow$ 任意值，$s \in \mathcal S$。如果有终止状态，$v_0\left(s_{\text{终止}}\right) \leftarrow 0$ 。} \\
&\text{2.（迭代）对于 $k \leftarrow 0, 1, 2, 3, \cdots,$ 迭代执行以下步骤：} \\
&\qquad \text{2.1  对于使用误差容忍度的情况，初始化本次迭代观测到的最大误差 } \delta \leftarrow 0\ ; \\
&\qquad \text{2.2  对于 $s \in \mathcal S$ 执行以下操作：} \\
&\qquad \qquad \text{2.2.1  计算新状态价值 } v_{\text{新}} \leftarrow \max_{a} \left(r\left(s,a\right) + \gamma \sum_{s'} p\left(s' \mid s, a\right) v\left(s'\right)\right) ; \\
&\qquad \qquad \text{2.2.2  对于使用容忍度误差的情况，更新本次迭代观测到的最大误差 } \delta \leftarrow \max \left(\;\delta,\; |v_{\text{新}} - v\left(s\right)|\;\right); \\
&\qquad \qquad \text{2.2.3  更新状态价值函数 } v\left(s\right) \leftarrow v_{\text{新}}\;; \\
&\qquad \text{2.3  若满足容忍度误差（即 $\delta \lt \delta_{max}$ ），或达到最大迭代次数（即 $k=k_{max}$），则跳出循环。} \\
&\text{3.（策略）根据价值函数输出确定性策略：} \\
\end{split} \\
\pi \left(s\right) \leftarrow \underset{a}{\arg\max} \left(r\left(s,a\right) + \gamma \sum_{s'} p\left(s' \mid s,a\right) v\left(s'\right)\right) 。\\
\begin{split}
&\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em} \\
\end{split}
\; \\ \; \\</script></div></div><h3><a name="四案例冰面滑行frozenlake-v0）" class="md-header-anchor"></a><span>四、案例：冰面滑行（FrozenLake-v0）</span></h3><p><span>使用 gym 库中的冰面滑行问题（FrozenLake-v0）作为案例分析，该环境也是一个有限 Markov 决策过程，具体环境和规则可查看</span><a href='https://github.com/openai/gym/blob/master/gym/envs/toy_text/frozen_lake.py' target='_blank' title='FrozenLake-v0'><span>源代码</span></a><span>。书上的源代码是使用算法 3-2 实现策略评估算法，使用算法 3-4 实现策略改进算法，使用算法 3-6 实现策略迭代算法来获取最优策略及其状态价值函数。</span></p><p><span>根据书上的源代码，将其改写成了一个类 </span><code>PolicyIteration</code><span> ，为了方便研究整个评估和改进过程，将动作价值函数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.226ex" height="2.71ex" viewBox="0 -832.7 2680.7 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E325-MJMATHI-71" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E325-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E325-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E325-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E325-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E325-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E325-MJMATHI-71" x="0" y="0"></use><use xlink:href="#E325-MJMAIN-28" x="460" y="0"></use><use xlink:href="#E325-MJMATHI-73" x="849" y="0"></use><use xlink:href="#E325-MJMAIN-2C" x="1318" y="0"></use><use xlink:href="#E325-MJMATHI-61" x="1762" y="0"></use><use xlink:href="#E325-MJMAIN-29" x="2291" y="0"></use></g></svg></span><script type="math/tex">q(s,a)</script><span> 也存储了下来，所以策略改进部分没有严格按算法实现，并且并综合修改了其他模块的部分语句，改写的代码如下：</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 43px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 39px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><span><span>​</span>x</span></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -39px; width: 39px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">class</span> <span class="cm-def">PolicyIteration</span>():</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">def</span> <span class="cm-def">__init__</span>(<span class="cm-variable-2">self</span>, <span class="cm-variable">env</span>, <span class="cm-variable">gamma</span>=<span class="cm-number">0.99</span>, <span class="cm-variable">tolerant</span>=<span class="cm-number">1e-6</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">env</span> = <span class="cm-variable">env</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">gamma</span> = <span class="cm-variable">gamma</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">tolerant</span> = <span class="cm-variable">tolerant</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">policy</span> = <span class="cm-variable">np</span>.<span class="cm-property">ones</span>((<span class="cm-variable">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>, <span class="cm-variable">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>)) <span class="cm-operator">/</span> <span class="cm-variable">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>(<span class="cm-variable">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">q_table</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>((<span class="cm-variable">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>, <span class="cm-variable">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">policy_is_optimal</span> = <span class="cm-keyword">False</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">def</span> <span class="cm-def">v2q</span>(<span class="cm-variable-2">self</span>, <span class="cm-variable">single_state</span>=<span class="cm-keyword">None</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">single_state</span> <span class="cm-keyword">is</span> <span class="cm-keyword">not</span> <span class="cm-keyword">None</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">q_line</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">action</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">proba</span>, <span class="cm-variable">next_state</span>, <span class="cm-variable">reward</span>, <span class="cm-variable">done</span> <span class="cm-keyword">in</span> <span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">P</span>[<span class="cm-variable">single_state</span>][<span class="cm-variable">action</span>]:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">q_line</span>[<span class="cm-variable">action</span>] += <span class="cm-variable">proba</span> <span class="cm-operator">*</span> (<span class="cm-variable">reward</span> <span class="cm-operator">+</span> <span class="cm-variable-2">self</span>.<span class="cm-property">gamma</span> <span class="cm-operator">*</span> <span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span>[<span class="cm-variable">next_state</span>] <span class="cm-operator">*</span> (<span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">done</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">return</span> <span class="cm-variable">q_line</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">state</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">q_table</span>[<span class="cm-variable">state</span>] = <span class="cm-variable-2">self</span>.<span class="cm-property">v2q</span>(<span class="cm-variable">state</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">def</span> <span class="cm-def">evaluate_policy</span>(<span class="cm-variable-2">self</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span>[:] = <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">while</span> <span class="cm-keyword">True</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">delta</span> = <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">state</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">v_new</span> = <span class="cm-builtin">sum</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">policy</span>[<span class="cm-variable">state</span>] <span class="cm-operator">*</span> <span class="cm-variable-2">self</span>.<span class="cm-property">v2q</span>(<span class="cm-variable">state</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">delta</span> = <span class="cm-builtin">max</span>(<span class="cm-variable">delta</span>, <span class="cm-builtin">abs</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span>[<span class="cm-variable">state</span>]<span class="cm-operator">-</span><span class="cm-variable">v_new</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span>[<span class="cm-variable">state</span>] = <span class="cm-variable">v_new</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">delta</span> <span class="cm-operator">&lt;</span> <span class="cm-variable-2">self</span>.<span class="cm-property">tolerant</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">break</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">def</span> <span class="cm-def">improve_policy</span>(<span class="cm-variable-2">self</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">v2q</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">actions</span> = <span class="cm-variable">np</span>.<span class="cm-property">argmax</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">q_table</span>, <span class="cm-variable">axis</span>=<span class="cm-number">1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">policy</span> = <span class="cm-variable">np</span>.<span class="cm-property">eye</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>, <span class="cm-error">\</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">37</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>)[<span class="cm-variable">actions</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">38</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> (<span class="cm-variable-2">self</span>.<span class="cm-property">policy</span> == <span class="cm-variable">policy</span>).<span class="cm-property">all</span>():</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">39</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">policy_is_optimal</span> = <span class="cm-keyword">True</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">40</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">41</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">policy</span> = <span class="cm-variable">policy</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">42</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">43</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">def</span> <span class="cm-def">iterate_policy</span>(<span class="cm-variable-2">self</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">44</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">while</span> <span class="cm-keyword">True</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">45</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">evaluate_policy</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">46</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">improve_policy</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">47</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable-2">self</span>.<span class="cm-property">policy_is_optimal</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">48</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">break</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1568px;"></div><div class="CodeMirror-gutters" style="height: 1568px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 38px;"></div></div></div></div></pre><p><span>对于该案例也可以使用价值迭代算法去求解最优策略，书上的源代码是使用算法 3-8 实现的价值迭代算法，同样的，也将其改写成了一个类 </span><code>ValueIteration</code><span> ，并且记录了动作价值函数  </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.226ex" height="2.71ex" viewBox="0 -832.7 2680.7 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E325-MJMATHI-71" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E325-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E325-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E325-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E325-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E325-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E325-MJMATHI-71" x="0" y="0"></use><use xlink:href="#E325-MJMAIN-28" x="460" y="0"></use><use xlink:href="#E325-MJMATHI-73" x="849" y="0"></use><use xlink:href="#E325-MJMAIN-2C" x="1318" y="0"></use><use xlink:href="#E325-MJMATHI-61" x="1762" y="0"></use><use xlink:href="#E325-MJMAIN-29" x="2291" y="0"></use></g></svg></span><script type="math/tex">q(s,a)</script><span> ，改写的代码如下：</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 43px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 39px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>32</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -39px; width: 39px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">class</span> <span class="cm-def">ValueIteration</span>():</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">def</span> <span class="cm-def">__init__</span>(<span class="cm-variable-2">self</span>, <span class="cm-variable">env</span>, <span class="cm-variable">gamma</span>=<span class="cm-number">0.99</span>, <span class="cm-variable">tolerant</span>=<span class="cm-number">1e-6</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">env</span> = <span class="cm-variable">env</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">gamma</span> = <span class="cm-variable">gamma</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">tolerant</span> = <span class="cm-variable">tolerant</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>(<span class="cm-variable">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">q_table</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>((<span class="cm-variable">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>, <span class="cm-variable">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">def</span> <span class="cm-def">v2q</span>(<span class="cm-variable-2">self</span>, <span class="cm-variable">single_state</span>=<span class="cm-keyword">None</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">single_state</span> <span class="cm-keyword">is</span> <span class="cm-keyword">not</span> <span class="cm-keyword">None</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">q_line</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">action</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">proba</span>, <span class="cm-variable">next_state</span>, <span class="cm-variable">reward</span>, <span class="cm-variable">done</span> <span class="cm-keyword">in</span> <span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">P</span>[<span class="cm-variable">single_state</span>][<span class="cm-variable">action</span>]:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">q_line</span>[<span class="cm-variable">action</span>] += <span class="cm-variable">proba</span> <span class="cm-operator">*</span> (<span class="cm-variable">reward</span> <span class="cm-operator">+</span> <span class="cm-variable-2">self</span>.<span class="cm-property">gamma</span> <span class="cm-operator">*</span> <span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span>[<span class="cm-variable">next_state</span>] <span class="cm-operator">*</span> (<span class="cm-number">1.0</span> <span class="cm-operator">-</span> <span class="cm-variable">done</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">return</span> <span class="cm-variable">q_line</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">state</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">q_table</span>[<span class="cm-variable">state</span>] = <span class="cm-variable-2">self</span>.<span class="cm-property">v2q</span>(<span class="cm-variable">state</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp;<span class="cm-keyword">def</span> <span class="cm-def">iterate_value</span>(<span class="cm-variable-2">self</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">while</span> <span class="cm-keyword">True</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">delta</span> = <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">for</span> <span class="cm-variable">state</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">v_max</span> = <span class="cm-builtin">max</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">v2q</span>(<span class="cm-variable">state</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">delta</span> = <span class="cm-builtin">max</span>(<span class="cm-variable">delta</span>, <span class="cm-builtin">abs</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span>[<span class="cm-variable">state</span>]<span class="cm-operator">-</span><span class="cm-variable">v_max</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">v_table</span>[<span class="cm-variable">state</span>] = <span class="cm-variable">v_max</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">if</span> <span class="cm-variable">delta</span> <span class="cm-operator">&lt;</span> <span class="cm-variable-2">self</span>.<span class="cm-property">tolerant</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-keyword">break</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">v2q</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable">actions</span> = <span class="cm-variable">np</span>.<span class="cm-property">argmax</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">q_table</span>, <span class="cm-variable">axis</span>=<span class="cm-number">1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 30px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">policy</span> = <span class="cm-variable">np</span>.<span class="cm-property">eye</span>(<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">observation_space</span>.<span class="cm-property">n</span>, <span class="cm-error">\</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -39px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 30px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="cm-variable-2">self</span>.<span class="cm-property">env</span>.<span class="cm-property">action_space</span>.<span class="cm-property">n</span>)[<span class="cm-variable">actions</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1056px;"></div><div class="CodeMirror-gutters" style="height: 1056px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 38px;"></div></div></div></div></pre></div>
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